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Chapter 12: Problem 22

The unit cell for \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) has hexagonal symmetry with lattice parameters \(a=0.4961 \mathrm{~nm}\) and \(c=1.360 \mathrm{~nm}\). If the density of this material is \(5.22 \mathrm{~g} / \mathrm{cm}^{3}\), calculate its atomic packing factor. For this computation assume ionic radii of \(0.062 \mathrm{~nm}\) and \(0.140 \mathrm{~nm}\), respectively, for \(\mathrm{Cr}^{3+}\) and \(\mathrm{O}^{2-}\).

Short Answer

Expert verified
Question: Calculate the atomic packing factor for the unit cell of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) which has hexagonal symmetry and given lattice parameters. Step 1: Calculate the volume of the unit cell. Step 2: Convert the volume to cm\(^3\). Step 3: Calculate the number of formula units of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) in the unit cell. Step 4: Calculate the volume occupied by the ions. Step 5: Calculate the atomic packing factor.

Step by step solution

01

Find the volume of the unit cell

Hexagonal unit cell volume is given by the formula: \(V = \frac{3\sqrt{3}}{2}a^2c\) Given \(a = 0.4961 \,\text{nm}\) and \(c = 1.360 \,\text{nm}\), we can find the volume. \(V = \frac{3\sqrt{3}}{2}(0.4961)^2(1.360)\)
02

Convert the volume to cm\(^3\)

Since the density is given in g/cm\(^3\), we need to convert the volume of the unit cell from nm\(^3\) to cm\(^3\): \(1 \,\text{nm} = 10^{-7} \,\text{cm}\) So, \(V_\text{cm} = V \times (10^{-7})^3\)
03

Calculate the number of formula units of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) in the unit cell

Using the given density, \(\rho = 5.22 \,\text{g/cm}^3\), we can find the mass of the unit cell: \(m = V_\text{cm} \times \rho\) Now, we need to find the number of formula units of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) in the unit cell. Since the molar mass of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) is \((2\times52) + (3\times16) = 152 \,\text{gm/mol}\), we can calculate the number of formula units using the mass of the unit cell: \(n_\text{FU} = \frac{m}{152} \times N_\text{A}\) Here, \(N_\text{A}\) is Avogadro's number.
04

Calculate the volume occupied by the ions

We have the ionic radii for \(\mathrm{Cr}^{3+}\) and \(\mathrm{O}^{2-}\) to be \(0.062 \,\text{nm}\) and \(0.140 \,\text{nm}\) respectively. Assuming the ions to be spheres, we can calculate their individual volumes: \(V_\text{Cr} = \frac{4\pi}{3}(0.062)^3 \times 10^{-21} \,\text{cm}^3\) \(V_\text{O} = \frac{4\pi}{3}(0.140)^3 \times 10^{-21} \,\text{cm}^3\) Since there are two \(\mathrm{Cr}^{3+}\) ions and three \(\mathrm{O}^{2-}\) ions in the formula unit of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\), the total volume occupied by the ions in one formula unit is: \(V_\text{FU} = 2V_\text{Cr} + 3V_\text{O}\)
05

Calculate the atomic packing factor

Finally, we can calculate the atomic packing factor which is the ratio of volume occupied by the ions to the total volume of the unit cell: \(APF = \frac{n_\text{FU} \times V_\text{FU}}{V_\text{cm}}\) Calculate the atomic packing factor using the values obtained in the previous steps.

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